Let f(x) = x² +2° (a) On what exact interval(s) is ƒ increasing? decreasing? Make it clear which answers are which, and justify your answers with appropriate supporting work. (b) Find all local maximum and local minimum values of f, giving x and y coordinates with each answer Make it clear which answers are which, and justify your answers with appropriate supporting work.
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
I need help with both of these questions
![## Problem 9
Let \( f(x) = \frac{2x}{x^2 + 5} \). Below, you are given the first and second derivatives of \( f \); you need not calculate them yourself.
\[
f'(x) = \frac{-2(x^2 - 5)}{(x^2 + 5)^2} \quad \text{and} \quad f''(x) = \frac{4x(x^2 - 15)}{(x^2 + 5)^3}.
\]
### Questions
(a) **On what intervals is \( f \) increasing? decreasing?**
Make it clear which answers are which, and justify your answer by including an appropriate sign chart as part of your work.
(b) **Find all local maximum and local minimum values of \( f \), giving both \( x \) and \( y \) coordinates in each case.**
Please make it clear which answers are which.
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### Explanation
- **First Derivative \( f'(x) \):** This determines where the function \( f(x) \) is increasing or decreasing. A positive derivative indicates that \( f(x) \) is increasing, while a negative derivative indicates it is decreasing.
- **Second Derivative \( f''(x) \):** This helps identify concavity and potential points of inflection. Positive values indicate the graph is concave up, while negative values indicate concave down.
- **Sign Chart:** Use a sign chart for \( f'(x) \) to find intervals of increase and decrease. Look for critical points where \( f'(x) = 0 \) or is undefined.
- **Local Extrema:** Critical points from the first derivative and test using the second derivative or sign changes to determine if they are maxima or minima.
Use the given derivatives to solve the problem based on the intervals where these expressions change sign.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3bdf0c0d-f4c2-4c5b-a144-37d2c655f378%2F4957861a-a0fa-459b-945d-71c11c1a8398%2Fi7v4o2l_processed.png&w=3840&q=75)
Differentiate with respect to x
critical points, f'(x)=0
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